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REVIEW 2 major objections 5 minor 45 references

High-precision minmax solution of the two-center Dirac equation

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The two-center Dirac equation is solved by the minmax finite-element method to benchmark accuracy, with estimated fractional uncertainties of about 1e-23 in the relativistic shift for H2+ and 1e-21 for Th2^179+.

desk verdict H2+ benchmark is solid and agrees with independent work to 7e-30, but the Th2^179+ uncertainty claim is overstated by roughly an order of magnitude and needs revision. read the letter →

arxiv 2411.12427 v1 pith:BSVV3LFR submitted 2024-11-19 quant-ph physics.atm-clus

Ossama Kullie This is my paper · ORCID
classification quant-phphysics.atm-clus MSC 65N3081Q0581V45 PACS 31.15.-p31.30.J02.70.Dh
keywords two-centerDiracequationfiniteelementmethodminmaxprinciplerelativisticshiftH2+molecularionheavyquasi-molecularTh2^179+Coulombsingularityhigh-precisionbenchmark
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper reports a benchmark-quality solution of the two-center Dirac equation, the relativistic one-electron problem of two Coulomb centers that models molecular hydrogen ions and heavy quasi-molecular ions. The method combines the minmax variational principle, which projects out the negative-energy (positronic) continuum, with a high-order finite-element discretization and a singular coordinate transformation that regularizes the point-nucleus Coulomb singularity. For ${\rm H}_2^+$ at $R=2$, the final relativistic ground-state energy is $-1.10264158103257716411812499995$ atomic units, and for ${\rm Th}_2^{179+}$ at $R=2/90$ it is $-9504.756648434009500737$ atomic units; the corresponding relativistic shifts carry estimated fractional uncertainties of about $10^{-23}$ and $10^{-21}$. If these values are correct, they provide reference energies for QED corrections and bound-electron g-factor calculations in molecular ions.

What carries the argument

The load-bearing machinery is the minmax energy functional, where the small spinor component is eliminated and the large-component equation becomes a nonlinear eigenvalue problem solved iteratively. The two-dimensional finite-element calculation uses complete polynomial shape functions of order $p=10$ in prolate spheroidal coordinates $(\xi,\eta)$, together with the singular coordinate transformation of Eq. (A1), whose order $\nu$ controls the point density near the nuclei and regularizes the Coulomb singularity of the form $r^{-1+\gamma_{l,\kappa}}$. The relativistic shift is computed as $E_{\rm rel}-E_{\rm nrel}$ on identical grids, which cancels most of the smooth discretization error, and a power-law extrapolation over the grid sequence with the fitted convergence order $q$ as the leading error exponent produces the final energies. This combination of projection, singularity regularization, and error cancellation is what carries the claim.

What would settle it

Refine both systems beyond the densest grids used here (1800 elements, 90601 points), for ${\rm Th}_2^{179+}$ also with a higher $\nu$ value; if the extrapolated energies move by more than about $10^{-28}$ au for ${\rm H}_2^+$ or $10^{-18}$ au for ${\rm Th}_2^{179+}$, the claimed benchmark uncertainties are falsified.

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Extended reading notes

Core claim

The central claim is that the minmax finite-element method solves the two-center Dirac equation with far higher precision than earlier relativistic molecular calculations: absolute energy accuracy near $10^{-28}$ au for ${\rm H}_2^+$ and $10^{-18}$ au for ${\rm Th}_2^{179+}$, with the relativistic shift (the difference between Dirac and Schrödinger energies) even more accurate because the non-relativistic and relativistic computations on the same grids share the same systematic errors. The energies converge from above, no spurious states appear, and extrapolation over grid refinements with fitted convergence orders $q \approx 9.7$ and $q \approx 9.3$ yields the final values. The results agree with a recent independent high-precision calculation to about $7\times10^{-30}$ au for ${\rm H}_2^+$ and $5\times10^{-18}$ au for ${\rm Th}_2^{179+}$, and the paper argues that the minmax solution is cleaner in the strongly relativistic regime because it is free from positronic contamination.

Load-bearing premise

The uncertainty estimates assume that the extrapolated finite-element energies converge to the exact eigenvalue as a single power law with the fitted order $q$, and that the singular coordinate transformation with $\nu=10$ fully controls the Coulomb singularity for ${\rm Th}_2^{179+}$; the paper itself notes that for high $Z$ the singularity error outweighs the convergence order.

Editorial extensions

If this is right

  • The ${\rm H}_2^+$ ground-state energy at $R=2$ becomes a fixed reference with about 28 decimal digits, useful for benchmarking other relativistic methods and for QED one-loop self-energy calculations.
  • The ${\rm Th}_2^{179+}$ energy at $R=2/90$ improves on earlier heavy quasi-molecular results by many orders and provides a point-nucleus benchmark for strongly relativistic two-center systems.
  • The combination of minmax projection and singular-coordinate FEM should transfer to other internuclear distances and angular momentum channels, giving controlled convergence from above.
  • The error-cancelled relativistic shift is the quantity entering g-factor and radiative-correction calculations in molecular hydrogen ions, so the improved precision directly tightens theory-experiment comparisons.
  • The computational cost (quadruple precision, grids up to 90601 points) is moderate enough that the paper expects extension to multi-electron systems in Hartree-Fock or DFT frameworks.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • One testable extension is to go beyond the densest grids used here (1800 elements, 90601 points) or to raise $\nu$ for ${\rm Th}_2^{179+}$; if the extrapolated energy moves by more than the claimed $10^{-18}$ au, the uncertainty estimate would need revision.
  • The paper's uncertainty model assumes a single leading power-law error; if higher-order or oscillatory terms contribute at the last digits, the extrapolation could be biased, which a deliberately varied grid-sequence study could reveal.
  • Applying the same method at several internuclear distances $R$ would turn the single-point benchmark into an interpolation table for molecular spectroscopy and QED calculations.
  • Because the point-nucleus model is used, the heavy-ion value is a pure mathematical benchmark; adding finite-nuclear-size corrections would shift it at levels likely larger than the claimed numerical uncertainty, so direct experimental comparison needs that separate step.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. This paper computes the ground-state (1σg) solutions of the two-center Dirac equation for H2+ (R = 2) and Th2^179+ (R = 2/90) using the minmax functional and a p = 10 finite-element discretization in prolate spheroidal coordinates, with a singular coordinate transformation of order ν to regularize the Coulomb singularities. Systematic grid sequences (Tables I and III) yield Erel, Enrel, and the relativistic shift; power-law extrapolation with fitted orders q ≈ 9.7 and q ≈ 9.3 produces the final values in Table V, with claimed fractional uncertainties of ~10^-23 and ~10^-21 in the two shifts. The results agree with the independent calculation of Nogueira and Karr [19] to ~7×10^-30 (H2+) and ~5×10^-18 (Th2^179+). Additional cross-checks vary the domain size Dmax (Tables II, IV), the number of integration points per element (Fig. 2b), and the transformation order ν.

Significance. If the results hold at the stated precision, these are benchmark-quality reference values for two-center relativistic one-electron systems, of use for QED corrections, g-factor studies, and method benchmarking in molecular relativistic quantum chemistry. The manuscript's strengths are concrete: systematic grid-convergence tables (Tables I and III), cross-checks against domain size (Tables II and IV), integration-point count (Fig. 2b), and transformation order; verification against an independent calculation [19]; and a minmax formulation whose raw FEM values converge from above without spurious states. The H2+ claim is fully supported by the ~7×10^-30 agreement with [19]. The Th2^179+ uncertainty estimate is not supported by the manuscript's own data (Major Comment 1), and the α-uncertainty propagation is misstated (Major Comment 2); both issues are localized and fixable within the scope of the paper.

major comments (2)
  1. [Sec. III; Tables III–V; Fig. 2] The claimed fractional uncertainty of ~10^-21 in the Th2^179+ relativistic shift (absolute ~5.7×10^-19 au in a shift of -573.42 au) is not supported by the manuscript's own data. In Table IV, the extrapolated Erel values scatter over Dmax ∈ [0.30, 0.40] by ~3.6×10^-18 au, and the independent calculation [19] differs from the Table V value by 5×10^-18 au in Erel, an order of magnitude above the claimed shift uncertainty. The two internally consistent routes to the final energy also disagree: the direct ν = 10 extrapolation (Table III, -9504.756648434009500748) and the ν = 2 nonrelativistic value plus the extrapolated shift (Table V, -9504.756648434009500737) differ by ~1.1×10^-17 au, which exceeds the stated ~10^-18 energy uncertainty by an order of magnitude. The extrapolation assumes a single power law with q ≈ 9.3 fitted to the last grid points (Fig. 2), and the text itself states that for high Z 'the singularity error outweighs the convergence order' and that error cancellation in the shift is less efficient (Sec. III). The sentence that the discrepancy scatters around ≲10^-18 also conflicts with the reported 5×10^-18 difference from [19]. The authors should provide an error budget that accounts for the Table IV scatter, the route dependence, and the comparison with [19], or reduce the claimed precision to a level consistent with these data.
  2. [Sec. III (α-uncertainty paragraph)] The propagation of the CODATA 2018 α uncertainty is misstated. With α^-1 = 137.035999084(21), the relative uncertainty is δα/α ≈ 1.5×10^-10. Since the relativistic shift scales as α^2 at fixed Z at leading order, δ(ΔE)/ΔE ≈ 2δα/α ≈ 3×10^-10, giving δ(ΔE) ≈ 2×10^-15 au for H2+ and ≈ 1.7×10^-7 au for Th2^179+. This contradicts the text's claim of 'uncertainty of order 10^-20 in the obtained energies' and the statement that the α-uncertainty is smaller for Th2^179+ than for H2+. In fact the α-induced fractional uncertainty (~3×10^-10) exceeds the claimed computational fractional uncertainties (10^-23 and 10^-21) by many orders of magnitude for both systems. The authors should correct this analysis and state explicitly that the quoted fractional uncertainties are computational, valid at the fixed CODATA 2018 α value, rather than uncertainties of the physical values.
minor comments (5)
  1. [Table III] In Table III, the element counts '200/1020' and '1152/5808' appear to be missing a digit; they should read 200/10201 and 1152/58081 to match the grid sequence used for H2+ in Table I.
  2. [Table V] The provenance of each Table V entry should be specified: the tabulated Erel (-9504.756648434009500737) is not exactly the sum of the tabulated Enrel and the shift (-8931.3371374090663382226 - 573.4195110249431625138 = -9504.7566484340095007364), so the reader cannot reproduce the final values without knowing the parameter sets (ν, Dmax) used for each column.
  3. [Secs. II–III; Fig. 2; Conclusion] There are several typos and grammatical slips: 'minamx' (Sec. II), 'the the subspace of positronic states' (Sec. II), an unbalanced parenthesis in 'Z < 1/α < 137.036..)' (Sec. II), 'whrer' (Fig. 2 caption), 'blow the precision' for 'below the precision' (Sec. III), and missing spaces in the Conclusion paragraph ('Weshowedsystematicallyaccuratevaluesbyinvestigationoftheconvergence').
  4. [Figs. 1–2] The paper reports the slopes of the linear fits in Figs. 1 and 2 but does not state how many grid points enter each fit or how the extrapolated value changes when the fit window is varied; specifying the fit range and the sensitivity to it would support reproducibility of the extrapolation step.
  5. [Table V] The label 'Rel. eff.' in Table V is not defined; the text uses 'relativistic shift', 'relativistic effect', and 'Rel. eff.' interchangeably, and the notation should be unified.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the central benchmark values are validated against the independent calculation of Nogueira and Karr [19] and against the non-relativistic limit; self-citations concern method provenance and are not load-bearing.

full rationale

The derivation chain is self-contained as a numerical computation. The paper solves the FEM-discretized minmax two-center Dirac equation (Eq. 4) with an iterative expansion (Eq. 5), then extrapolates a grid sequence. The convergence order q entering the extrapolation is fitted to the same error sequence, but this is a standard convergence-acceleration step: the reported energies are the FEM values themselves, and q only models the rate at which they approach the exact eigenvalue. No fitted parameter is set to force agreement with an external target. The final claim is independently grounded: 'Looking at table V on finds an excellent agreement with the result of Nogueira et. al. [19]. The discrepancies are ∼ 7.10−30, 5. 10−18.' The method is attributed partly to the author's earlier work ('we apply the method developed in earlier studies [6, 7]'), but the minmax principle itself is the external mathematical characterization of the Dirac eigenvalues [1], and the numerical implementation is testable against [19] and the known non-relativistic limit (c → ∞). This is not a self-citation chain used to forbid alternatives. The Thorium uncertainty concern raised by the paper's own statement that 'the singularity error outweighs the convergence order' is a robustness/correctness issue about error bounds, not circularity: the quoted discrepancies with [19] are external evidence, and overstating an uncertainty does not make the derivation equivalent to its inputs. No self-definitional, fitted-input-called-prediction, or uniqueness-imported-from-authors pattern is present.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

No new physics entities are introduced. The calculation loads its epistemology onto (i) the minmax reduction of the Dirac spectrum, (ii) the point-nucleus model, (iii) a hand-tuned singular coordinate transformation, and (iv) a power-law extrapolation whose order is fitted from the data. The free parameters are all numerical discretization choices; their convergence is partially verified by grid sequences and Dmax scans, but the resulting uncertainty bars are not derived from a formal error analysis.

free parameters (5)
  • nu (singular coordinate transformation order) = 8 for H2+, 10 for Th2^179+
    Chosen by hand to concentrate grid points near the Coulomb singularity; higher nu improves convergence for relativistic calculations (Section III, Figs. 1-2).
  • Dmax (domain size) = 40 au for H2+, 0.35 au for Th2^179+
    Chosen by hand as optimal; Tables II and IV show weak dependence on Dmax, but it affects the error cancellation between relativistic and non-relativistic energies.
  • FEM polynomial order p = 10
    Set to p=10 in all reported calculations; determines the FEM convergence order.
  • Integration points per element nI = 25 (20 for tests)
    Number of integration points per triangular element; chosen for accuracy, with nI=20-25 tested.
  • Extrapolation order q = 9.7 for H2+, 9.3 for Th2^179+
    Fitted from linear regions of log-log error plots; used to extrapolate energy to infinite grid size. This is a fitted parameter of the error model, not a physical constant.
assumptions (4)
  • domain assumption The minmax variational principle yields the electronic spectrum of the Dirac operator for point-nucleus Coulomb potentials with Z < 1/alpha.
    Invoked in Section II to define the method; relies on theorems of Dolbeault, Esteban, Sere, and Griesmer et al. [1-4].
  • domain assumption Point-nucleus model with Z < 137 is used; finite nuclear size effects are neglected.
    Stateed in Section III near the end; the paper works with point charges, which is standard for method benchmarking.
  • standard math The singular coordinate transformation (eq. A1) with parameter nu regularizes the Coulomb singularity so that FEM converges at order p.
    Used in Appendix A; the transformation is designed to increase point density near nuclei, following previous works [11,16].
  • ad hoc to paper The FEM approximation error follows a power law E(N) = E_exact + C N^(-q), used for Richardson extrapolation.
    The extrapolation in Tables I and III assumes a single power-law error model; the order q is fitted from the same data, which is not a rigorous error bound.

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Cite this review

Pith. "Pith review of High-precision minmax solution of the two-center Dirac equation." pith.science (2026). https://pith.science/paper/BSVV3LFR

@misc{pith2026241112427,
  author       = {Pith},
  title        = {Pith review of: High-precision minmax solution of the two-center Dirac equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BSVV3LFR}},
  note         = {Machine review of arXiv:2411.12427}
}
abstract

We present a high-precision solution of Dirac equation by numerically solving the minmax two-center Dirac equation with the finite element method (FEM). The minmax FEM provide a highly accurate benchmark result for systems with light or heavy atomic nuclear charge $Z$. A result is shown for the molecular ion ${\rm H}_2^+$ and the heavy quasi-molecular ion ${\rm Th}_2^{179+}$, with estimated fractional uncertainties of $\sim 10^{-23}$ and $\sim 10^{-21}$, respectively. The result of the minmax-FEM high-precision of the solution of the two-center Dirac equation, allows solid control over the required accuracy level and is promising for the application and extension of our method.

Figures

Figures reproduced from arXiv: 2411.12427 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

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